Bidodecateron

Bidodecateron
File:Bidodecateron.png (OFF file)
Rank	5
Type	Noble
Notation
Bowers style acronym	Bidot
Coxeter diagram	o3o3m3o3o ()
Elements
Tera	20 triangular duotegums
Cells	90 tetragonal disphenoids
Faces	120 isosceles triangles
Edges	30+30
Vertices	12
Vertex figure	Joined pentachoron
Measures (based on 2 5-simplices of edge length 1)
Edge lengths	Lacing edges (30): ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Base edges (30): 1
Circumradius	${\frac {\sqrt {15}}{6}}\approx 0.64550$
Inradius	${\frac {\sqrt {6}}{6}}\approx 0.40825$
Diteral angle	$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Height	${\frac {\sqrt {6}}{3}}\approx 0.81650$
Central density	1
Related polytopes
Army	Bidot
Regiment	Bidot
Dual	Dodecateron
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5×2, order 1440
Convex	Yes
Nature	Tame

The bidodecateron or bidot, also known as the triangular-duotegmatic icosateron or triangular duotegmatic alterprism, is a convex noble polyteron with 20 identical triangular duotegums as facets. 10 facets join at each vertex, with the vertex figure being a joined pentachoron. It can be obtained as the convex hull of a hexateron and its central inversion (or, equivalently, its dual). It is also the triangular member of an infinite series of isogonal duotegmatic alterprisms.

The ratio between the longest and shortest edges is $1:{\frac {\sqrt {6}}{2}}\approx 1:1.22474$.

Vertex coordinates

The vertices of a bidodecateron, based on two hexatera of edge length 1, centered at the origin, are given by:

• $\pm \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}}\right)$,
• $\pm \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}}\right)$,
• $\pm \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}}\right)$,
• $\pm \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,\pm {\frac {\sqrt {15}}{6}}\right)$.

External links

• Klitzing, Richard. "bidot".